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arXiv:cond-mat/0606291 · Physical Review B

Phase transition and phase diagram at a general filling in the spinless one-dimensional Holstein model

Sanjoy Datta, Sudhakar Yarlagadda

Published Jan 23, 2007• 7 citations
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Abstract

We derive the Luttinger liquid to charge density wave (CDW) transition condition, exact to second order in a blocked perturbative approach, for the spinless one-dimensional Holstein model in the adiabatic regime. The small parameter is the ratio of the electron-phonon coupling $g$ and the adiabaticity parameter $t∕{\ensuremath{\omega}}_{0}$. Here we correct the mean-field criterion for Peierls instability, by replacing the static noninteracting susceptibility at twice the Fermi momentum ${\ensuremath{\chi}}_{0}(2{k}_{F},0)$ with the dynamic one, and get $1+2{g}^{2}{\ensuremath{\omega}}_{0}{\ensuremath{\chi}}_{0}(2{k}_{F},{\ensuremath{\omega}}_{0})=0$. At non-half-filling, we present the phase diagram showing the surprising result that the CDW occurs in a more restricted region of the two-parameter (${g}^{2}{\ensuremath{\omega}}_{0}∕t$ and $t∕{\ensuremath{\omega}}_{0}$) space than at half-filling.

DOI
10.1103/physrevb.75.035124

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We derive the Luttinger liquid to charge density wave (CDW) transition condition, exact to second order in a blocked perturbative approach, for the spinless one-dimensional Holstein model in the adiabatic regime. The small parameter is the ratio of the electron-phonon coupling ggg and the adiabaticity parameter t∕\ensuremathω0t∕{\ensuremath{\omega}}_{0}t∕\ensuremathω0​. Here we correct the mean-field criterion for Peierls instability, by replacing the static noninteracting susceptibility at twice the Fermi momentum \ensuremathχ0(2kF,0){\ensuremath{\chi}}_{0}(2{k}_{F},0)\ensuremathχ0​(2kF​,0) with the dynamic one, and get 1+2g2\ensuremathω0\ensuremathχ0(2kF,\ensuremathω0)=01+2{g}^{2}{\ensuremath{\omega}}_{0}{\ensuremath{\chi}}_{0}(2{k}_{F},{\ensuremath{\omega}}_{0})=01+2g2\ensuremathω0​\ensuremathχ0​(2kF​,\ensuremathω0​)=0. At non-half-filling, we present the phase diagram showing the surprising result that the CDW occurs in a more restricted region of the two-parameter (g2\ensuremathω0∕t{g}^{2}{\ensuremath{\omega}}_{0}∕tg2\ensuremathω0​∕t and t∕\ensuremathω0t∕{\ensuremath{\omega}}_{0}t∕\ensuremathω0​) space than at half-filling.
Electron- and phonon spectral functions of the one-dimensional, spinless-fermion Holstein model at half filling are calculated in the four distinct regimes of the phase diagram, corresponding to an attractive or repulsive Luttinger liquid at weak electron-phonon coupling, and a band- or polaronic insulator at strong coupling. The results obtained by means of kernel polynomial and systematic cluster approaches reveal substantially different physics in these regimes and further indicate that the size of the phonon frequency significantly affects the nature of the quantum Peierls phase transition, the latter being either of the soft-mode or central-peak type. The generic features observed are relevant to several classes of low-dimensional materials.
The effects of quantum lattice fluctuations on the Peierls transition are studied within the one-dimensional Holstein molecular-crystal model by means of exact diagonalization methods. Applying a very efficient variational Lanczos technique, the ground-state phase diagram is obtained in excellent agreement with predictions of recent density matrix renormalization-group calculations. The transition to the charge-density-wave regime is signaled by a strong increase in the charge structure factor. In the metallic regime, the nonuniversal Luttinger liquid parameters (charge velocity and coupling constant) are deduced from a finite-size scaling analysis. The variational results are supported by a complete numerical solution of the quantum phonon Holstein model on small clusters, which is based on a well-controlled phonon Hilbert space truncation procedure. The metallic and charge-density-wave phases are characterized by significant differences in the calculated optical-absorption spectra.
We study the Holstein model of spinless fermions, which at half filling exhibits a quantum phase transition from a metallic Tomonaga-Luttinger liquid phase to an insulating charge-density-wave (CDW) phase at a critical electron-phonon coupling strength. In our work, we focus on the real-time evolution starting from two different types of initial states that are CDW ordered: (i) ideal CDW states with and without additional phonons in the system and (ii) correlated ground states in the CDW phase. We identify the mechanism for CDW melting in the ensuing real-time dynamics and show that it strongly depends on the type of initial state. We focus on the far-from-equilibrium regime and emphasize the role of electron-phonon coupling rather than dominant electronic correlations, thus complementing a previous study of photoinduced CDW melting [H. Hashimoto and S. Ishihara, Phys. Rev. B 96, 035154 (2017)]. The numerical simulations are performed by means of matrix-product-state based methods with a local basis optimization (LBO). Within these techniques, one rotates the local (bosonic) Hilbert spaces adaptively into an optimized basis that can then be truncated while still maintaining a high precision. In this work, we extend the time-evolving block decimation (TEBD) algorithm with LBO, previously applied to single-polaron dynamics, to a half-filled system. We demonstrate that in some parameter regimes, a conventional TEBD method without LBO would fail. Furthermore, we introduce and use a ground-state density-matrix renormalization group method for electron-phonon systems using local basis optimization. In our examples, we account for up to Mph=40{M}_{\mathrm{ph}}=40Mph​=40 bare phonons per site by working with O(10)O(10)O(10) optimal phonon modes.
We reexamine the nature of the metallic phase of the one-dimensional half-filled Holstein model of spinless fermions. To this end, we determine the Tomonaga-Luttinger–liquid correlation parameter K ρ by large-scale density matrix renormalisation group (DMRG) calculations, exploiting i) the leading-order scaling relations between the ground-state energy and the single-particle excitation gap and ii) the static charge structure factor in the long-wavelength limit. While both approaches give almost identical results for intermediate-to-large phonon frequencies, we find contrasting behaviour in the adiabatic regime: i) K ρ >1 (attractive) vs. ii) K ρ < 1 (repulsive). The latter result for the correlation exponent is corroborated by data obtained for the momentum distribution function n ( k ), which puts the existence of an attractive metallic state in the spinless fermion Holstein model into question. We conclude that the scaling relation must be modified in the presence of electron-phonon interactions with noticeable retardation.
We present a numerical study of the Hubbard-Holstein model in one dimension at half filling, including finite-frequency quantum phonons. At half filling, the effects of the electron-phonon and electron-electron interactions compete with the Holstein phonon coupling acting as an effective negative Hubbard on-site interaction U that promotes on-site electron pairs and a Peierls charge-density wave state. Most previous work on this model has assumed that only Peierls or Mott phases are possible at half filling. However, there has been speculation that a third metallic phase exists between the Peierls and Mott phases. We confirm the intermediate phase, and show that the Luttinger liquid correlation exponent K(rho) >1 in this region, indicating dominant superconducting pair correlations. We explore the full phase diagram as a function of Hubbard U, phonon coupling constant, and phonon frequency.
The ground-state phase diagram of the half filled one-dimensional Holstein-Hubbard model contains a charge-density-wave (CDW) phase, driven by the electron-phonon $(e\text{\ensuremath{-}}ph)$ coupling, and a spin-density-wave (SDW) phase, driven by the on-site electron-electron $(e\text{\ensuremath{-}}e)$ repulsion. Recently, the existence of a third phase, which is metallic and lies in a finite region of parameter space between these two gapped phases, has been claimed. We study this claim using a renormalization-group method for interacting electrons that has been extended to include also $e\text{\ensuremath{-}}ph$ couplings. Our method [Tsai et al., Phys. Rev. B 72, 054531 (2005); Philos. Mag. B 86, 2631 (2006)] treats $e\text{\ensuremath{-}}e$ and $e\text{\ensuremath{-}}ph$ interactions on an equal footing and takes retardation effects fully into account. We find a direct transition between the SDW and CDW states. We study the effects of retardation, which are particularly important near the transition, and find that umklapp processes at finite frequencies drive the CDW instability close to the transition.